8. Let W be a subspace of a finite dimensional vector space V. Prove thatthere is a basis B for V such that there is a subset C of B that is a basisfor W. (Hint: Start with finding C. Use it to find B. Notice that theproblem does not say "Prove that for every basis B..")

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Answered: 8. Let W be a subspace of a finite…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3) Suppose that U and W are subspaces of a vector space V with V = U OW. Suppose also that u1, ., um is a basis of U amd w1,., Wn is a basis of W. Prove U1, .., Um, W1, .. , Wn is a basis of V. 4) Show that the subspaces of R? are precisely {0}, R², and all lines in R² through the
4. Let S = {v,,v2, V3, V4, V5} where, 3 1 2 3 3 ,V2 9. ,V3 2 la -1 4 Find a basis for the space span(S) which is a subset of S. Then express each vector that is not in the basis as a linear combination of the basis vectors.
Which of the following is a basis for the vector space R3?
Let A = {d1, d,}, B= {B, 6} be two bases of a vector space V. Suppose that 5 b1+7b; Let also 1 Then the change-of-coordinates matrix PA-B from the basis A to the basis B and [x]a are given by:
please prove 3 and 4
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7. Let V be a vector space and suppose B = {u, v} is a basis for V. Prove that B' = {ku, v} is also a basis for V for every nonzero scalar k. (Hint: You must prove that B'is linearly independent and that it spans V).
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8. Let W be a subspace of a finite dimensional vector space V. Prove that there is a basis B for V such that there is a subset C of B that is a basis for W. (Hint: Start with finding C. Use it to find B. Notice that the problem does not say "Prove that for every basis B.")